The Definition of probability:

It

is a branch of mathematics that is used in calculating likelihood of the

occurrence of a certain event. Probability is quantified as a number between 0

and 1. For example, if the probability of an event to occur is 0, thus, this is

expressing that the event is impossible to happen and if the probability of an

event to occur is 1, thus, this is expressing that the event will certainly

happen.

Probability Theorems:

Theorem #1. P(A) = 1 ? P(A’).

Theorem #2. P(Ø) = 0.

Theorem

#3. If events A and B are such

that A ? B,

then P(A) ? P(B)

Theorem #4. P(A) ? 1.

Theorem

#5. For any two events A and B, P(A ? B)

= P(A) + P(B) ? P(A ? B).

Types of Random Variables:

A

random variable is the variable whose all possible values are numerical outcomes

of a random phenomenon where only one variable is associated to each event in

the experiment.

As

an example:

When

tossing a coin the sample space will be {H,T}. These two outcomes might be

indicated by the variable X =1 if the outcome is H, X=0 if the outcome is T

As

a second example: if a dice is rolled and the face of the dice is seen or

observed. The sample space is {1,2,3,4,5,6}. The variable X here can represent

the number on the face which means here that it is equal to the sample space

{1,2,3,4,5,6}.

The

random variable that takes countable number of values is called discrete.

As

an example of discrete variables:

We

can the X from the previous two examples.

Continuous random variables:

From

the name these random variables assume the value corresponding to any value in

an interval. As an example:

The

amount of the time (X) that a student takes to finish an exam (if the total

time of the exam is one hour) 0 ?

X ? 60.

Types of probability distributions:

The probability

distribution of a discrete random variable can be done as a table, graph, or

formula that will be specific the probability connected each possible value

that the random variable can be.

There are some

requirements that must be present as to distribute the discrete random variable

correctly:

1. P(x) ? 0,

for all values of x.

2. ?p(x) = 1

Where the

summation of p(x) is over all possible values of x.

The mean: the sum of all the values divided by their number.

The variance of

a random variable:

The standard deviation of a random variable:

the square root of the variance.

The Binomial

distribution:

Many of the

experiments that we do or make could result in two possible outcomes as

Yes-No,Pass-Fail,etc.. As for example, when tossing a coin there are two

possible outcomes which are Tail or Heads. Random variables possessing these characteristics

are called binomial random variables.

These variables

have certain characteristics:

1. The

experiment consists of n identical trials.

2. There are

only two possible outcomes on each trial. We will denote one outcome by S (for

Success) and the other by F (for Failure).

3. The

probability of S remains the same from trial to trial. This probability is

denoted by p, and the probability of F is denoted by q = 1 ? p.

4. The trials

are independent.

5. The binomial

random variable X is the number of S’s in n trials.

The Poisson distribution:

When an event

happens randomly through time. For example, the occurring of hurricanes when

letting X be the number of times that the hurricanes occurring in an interval

of time. Then the following conditions must be accomplished in order to achieve

the Poisson distribution.

Characteristics

of a Poisson Random Variable 1.

The events occur at a constant average rate of ? per unit time.

2. Occurrences

are independent of one another.

3. More than

one occurrence cannot happen at the same time.

The probability

distributions for continuous random variables:

The probability

distribution of the continuous random variable is achieved by having it’s

density function. The following properties are achieved when the density

function F(X) with domain (a,b) :